Improving A*OMP: Theoretical and Empirical Analyses With a Novel Dynamic Cost Model

Best-first search has been recently utilized for compressed sensing (CS) by the A* orthogonal matching pursuit (A*OMP) algorithm. In this work, we concentrate on theoretical and empirical analyses of A*OMP. We present a restricted isometry property (RIP) based general condition for exact recovery of sparse signals via A*OMP. In addition, we develop online guarantees which promise improved recovery performance with the residue-based termination instead of the sparsity-based one. We demonstrate the recovery capabilities of A*OMP with extensive recovery simulations using the adaptive-multiplicative (AMul) cost model, which effectively compensates for the path length differences in the search tree. The presented results, involving phase transitions for different nonzero element distributions as well as recovery rates and average error, reveal not only the superior recovery accuracy of A*OMP, but also the improvements with the residue-based termination and the AMul cost model. Comparison of the run times indicate the speed up by the AMul cost model. We also demonstrate a hybrid of OMP and A?OMP to accelerate the search further. Finally, we run A*OMP on a sparse image to illustrate its recovery performance for more realistic coefcient distributions

Search-based methods for the sparse signal recovery problem in compressed

The sparse signal recovery, which appears not only in compressed sensing but also in other related problems such as sparse overcomplete representations, denoising, sparse learning, etc. has drawn a large attraction in the last decade. The literature contains a vast number of recovery methods, which have been analysed in theoretical and empirical aspects. This dissertation presents novel search-based sparse signal recovery methods. First, we discuss theoretical analysis of the orthogonal matching pursuit algorithm with more iterations than the number of nonzero elements of the underlying sparse signal. Second, best-first tree search is incorporated for sparse recovery by a novel method, whose tractability follows from the properly defined cost models and pruning techniques. The proposed method is evaluated by both theoretical and empirical analyses, which clearly emphasize the improvements in the recovery accuracy. Next, we introduce an iterative two stage thresholding algorithm, where the forward step adds a larger number of nonzero elements to the sparse representation than the backward one removes. The presented simulation results reveal not only the recovery abilities of the proposed method, but also illustrate optimal choices for the step sizes. Finally, we propose a new mixed integer linear programming formulation for sparse recovery. Due to the equivalency of this formulation to the original problem, the solution is guaranteed to be correct when it can be solved in reasonable time. The simulation results indicate that the solution can be found easily under some reasonable assumptions, especially for signals with constant amplitude nonzero elements

A comparison of termination criteria for A*OMP

Heuristic search has recently been utilized for compressed sensing signal recovery problem by the A* Orthogonal Matching Pursuit (A*OMP) algorithm. A*OMP employs A* search on a tree with an OMP-based evaluation of the branches, where the search is terminated when the desired path length is achieved. The algorithm employs effective pruning techniques and cost models which make the tree search practical. Here, we propose two important extensions of A*OMP: We first introduce a novel dynamic cost model that reduces the search time. Second, we modify the termination criterion by stopping the search when ℓ2 norm of the residue is small enough. Following the restricted isometry property, this termination criterion is more appropriate for our purposes. We demonstrate the improvements in terms of both reconstruction accuracy and computation times via a wide range of simulations

Optimal forward-backward pursuit for the sparse signal recovery problem (Seyrek işaret geri çatma problemi için optimal ileri-geri arayışı)

Forward-backward pursuit (FBP) is an iterative two stage thresholding method (TST) for sparse signal recovery. Due to the selection of more indices during the forward step than the ones pruned by the backward step, FBP iteratively enlarges the support estimate. With this structure, FBP does not necessitate the sparsity level to be known a priori in contrast to other TST algorithms such as subspace pursuit (SP) or compressive sampling matching pursuit. In this work, we address optimal selection of forward and backward step sizes for FBP. We analyse the empirical recovery performance of FBP with different step sizes via phase transitions. Moreover, we compare phase transitions of FBP with those of basis pursuit, SP and orthogonal matching pursuit

Multistage adaptive filtering for identification of page-oriented volume holographic memories

Page-oriented volume holographic memories (POVHM) is a quadratically nonlinear channel because of the intensity detection at its output. To combat the two-dimensional intersymbol interference in a high-capacity POVHM, equalization of the output intensity array requires identification of the quadratic and possibly spatially varying impulse response of the channel. Although conventional adaptive filtering schemes are devised for identification of linear channels, they also require the length of the impulse response to be known in advance. In this work, we develop multistage quadratic normalized least mean square (LMS) (MS-QNLMS) adaptive filtering and multistage Volterra normalized LMS (MS-VNLMS) filtering to estimate the channel under quadratic nonlinearity, which do not require the support or length of the impulse response to be known a priori. By employing extensive numerical experiments, we provide performance and convergence comparisons of the proposed schemes with respect to a true-order quadratic estimator. We also show that MS-QNLMS filtering has less computational complexity and converges faster and more robust to various channel parameters as compared to MS-VNLMS

Compressed sensing signal recovery via A* orthogonal matching pursuit

Reconstruction of sparse signals acquired in reduced dimensions requires the solution with minimum ℓ0 norm. As solving the ℓ0 minimization directly is unpractical, a number of algorithms have appeared for finding an indirect solution. A semi-greedy approach, A* Orthogonal Matching Pursuit (A*OMP), is proposed in [1] where the solution is searched on several paths of a search tree. Paths of the tree are evaluated and extended according to some cost function, for which novel dynamic auxiliary cost functions are suggested. This paper describes the A*OMP algorithm and the proposed cost functions briefly. The novel dynamic auxiliary cost functions are shown to provide improved results as compared to a conventional choice. Reconstruction performance is illustrated on both synthetically generated data and real images, which show that the proposed scheme outperforms well-known CS reconstruction methods

Sıkıştırmalı algılama işaret geri kazanma probleminde A* aramasının karmaşıklık ve doğruluk analizi (Analysis of accuracy and complexity of A* search for compressed sensing signal recovery)

A* Orthogonal Matching Pursuit (A*OMP) utilizes best-first search for recovery of sparse signals from reduced dimensions. It applies A* search on a tree whose branches are extended similar to Orthogonal Matching Pursuit (OMP). A*OMP makes a tractable tree search possible via effective complexity-accuracy trade-off parameters. Here, we concentrate on the effects of these parameters on the recovery performance. Via empirical comparison of complexity and performance, we demonstrate the effects of the search parameters on search size and recovery. We also compare A*OMP with well-known compressed sensing (CS) recovery techniques to reveal the improvement in the reconstruction

A mixed integer linear programming formulation for the sparse recovery problem in compressed sensing

We propose a new mixed integer linear programming (MILP) formulation of the sparse signal recovery problem in compressed sensing (CS). This formulation is obtained by introduction of an auxiliary binary vector, where ones locate the recovered nonzero indices. Joint optimization for finding this auxiliary vector together with the underlying sparse vector leads to the proposed MILP formulation. By addition of a few appropriate constraints, this problem can be solved by existing MILP solvers. In contrast to other methods, this formulation is not an approximation of the sparse optimization problem, but is its equivalent. Hence, its solution is exactly equal to the optimal solution of the original sparse recovery problem, once it is feasible. We demonstrate this by recovery simulations involving different sparse signal types. The proposed scheme improves recovery over the mainstream CS recovery methods especially when the underlying sparse signals have constant amplitude nonzero elements